4.7. TRIPLE INTEGRALS 385
4.7 Triple Integrals
4.7.1 Introduction
We define triple integrals over a 3-D solid in a way similar to the definition ofsingle integrals over an interval or double integrals over a 2-D region. We beginwith the simplest case, when the region of integration is a rectangular box. Wethen extend the definition to more general bounded solids (3-D region), andfocus on solids known as type 1, type2, and type 3 solids.
4.7.2 Triple Integrals Over a Rectangular Box
Consider the rectangular box
B ={
(x, y, z) ∈ R3 : a ≤ x ≤ b, c ≤ y ≤ d, r ≤ z ≤ s}
= [a, b]× [c, d]× [r, s]
We proceed as follows:
1. Divide B into sub-boxes. To achieve this, we do the following:
(a) We divide [a, b] into l subintervals [xi−1, xi] of equal width ∆x.
(b) We divide [c, d] into m subintervals [yj−1, yj ] of equal width ∆y.
(c) We divide [r, s] into n subintervals [zk−1, zk] of equal width ∆z.
(d) Then, B is divided into lmn sub-boxes Bijk (see figure 4.29) where
Bijk = [xi−1, xi]× [yj−1, yj ]× [zk−1, zk]
The volume of Bijk is ∆V = ∆x∆y∆z.
2. In each sub-box Bijk, we select a sample point(x∗ijk, y
∗ijk, z
∗ijk
).
3. We form the triple Riemann sum
l∑i=1
m∑j=1
n∑k=1
f(x∗ijk, y
∗ijk, z
∗ijk
)∆V
Definition 4.7.1 The triple integral of f over the box B is∫∫∫B
f (x, y, z) dV = liml,m,n→∞
l∑i=1
m∑j=1
n∑k=1
f(x∗ijk, y
∗ijk, z
∗ijk
)∆V
It can be shown that the triple integral always exists if f is continuous.
Computing such integrals will involve, as in the double integral case, iteratedintegrals. There is a version of Fubini’s theorem for triple integrals.
386 CHAPTER 4. MULTIPLE INTEGRALS
Figure 4.29: Rectangular Box Divided in Sub-boxes
4.7. TRIPLE INTEGRALS 387
Theorem 4.7.2 (Fubini’s Theorem for Triple Integrals) If f is continu-ous on the rectangular box B = [a, b]× [c, d]× [r, s] then∫∫∫
B
f (x, y, z) dV =
∫ b
a
∫ d
c
∫ s
r
f (x, y, z) dzdydx
Remark 4.7.3 Like in the double integral case, the order of integration can beswitched. There are six different iterated integrals which can be written. Weleave it to the reader to write these integrals.
Example 4.7.4 Evaluate∫∫∫
B
xyz2dV where B = [0, 1]× [−1, 2]× [0, 3].
By Fubini’s theorem, we have∫∫∫B
xyz2dV =
∫ 1
0
∫ 2
−1
∫ 3
0
xyz2dzdydx
The inner integrals is ∫ 3
0
xyz2dz =xyz3
3
∣∣∣∣30
= 9xy
Hence ∫ 2
−1
∫ 3
0
xyz2dzdy =
∫ 2
−19xydy
=9xy2
2
∣∣∣∣2−1
=36x
2− 9x
2
=27x
2
Finally ∫ 1
0
∫ 2
−1
∫ 3
0
xyz2dzdydx =
∫ 1
0
27x
2dx
=27x2
4
∣∣∣∣10
=27
4
388 CHAPTER 4. MULTIPLE INTEGRALS
4.7.3 Triple Integrals Over a General Bounded 3-D Solids
We then proceed as we did for double integrals.
• We consider a general 3-D bounded solid E.
• Because E is bounded, we enclose it in a box B.
• We define a function F to agree with f on E and to be 0 on B − E.
• We define ∫∫∫E
f (x, y, z) dV =
∫∫∫B
F (x, y, z) dV
• We focus our attention to continuous functions f and to certain simpletypes of solids.
4.7.4 Triple Integrals Over a Type 1 Solid
A solid E is said to be of type 1 if it lies between the graphs of two continuousfunctions in x and y that is
E ={
(x, y, z) ∈ R3 : (x, y) ∈ D,u1 (x, y) ≤ z ≤ u2 (x, y)}
Where D is the projection of E onto the xy-plane. as shown in figure 4.30.Usingan argument similar to the one used for double integrals, it can be shown that∫∫∫
E
f (x, y, z) dV =
∫∫D
[∫ u2(x,y)
u1(x,y)
f (x, y, z) dz
]dA
To integrate the inner integral, we hold x and y as constants and integrate withrespect to z.If D is a type 1 region in the plane, as described in the previous section (see
figure 4.31), then
E ={
(x, y, z) ∈ R3 : a ≤ x ≤ b, g1 (x) ≤ y ≤ g2 (x) , u1 (x, y) ≤ z ≤ u2 (x, y)}
and ∫∫∫E
f (x, y, z) dV =
∫ b
a
∫ g2(x)
g1(x)
∫ u2(x,y)
u1(x,y)
f (x, y, z) dzdydx
If on the other hand If D is a type 2 region in the plane, as described in theprevious section (see figure 4.32), then
E ={
(x, y, z) ∈ R3 : c ≤ y ≤ d, h1 (y) ≤ x ≤ h2 (y) , u1 (x, y) ≤ z ≤ u2 (x, y)}
and ∫∫∫E
f (x, y, z) dV =
∫ d
c
∫ h2(y)
h1(y)
∫ u2(x,y)
u1(x,y)
f (x, y, z) dzdxdy
4.7. TRIPLE INTEGRALS 389
Figure 4.30: A Type 1 Solid
390 CHAPTER 4. MULTIPLE INTEGRALS
Figure 4.31: A Type 1 Solid
4.7. TRIPLE INTEGRALS 391
Figure 4.32: A Type 1 Solid
392 CHAPTER 4. MULTIPLE INTEGRALS
Figure 4.33: Solid E
Example 4.7.5 Evaluate∫∫∫
E
zdV where E is the solid tetrahedron bounded
by the four planes x = 0, y = 0, z = 0, and x+ y+ z = 1. Pictures of E and Dare shown on the next slide. The solid E and the region D are shown on figures4.33 and 4.34.Here, the solid E and the region D are shown. When working on a problem, itis wise to draw them. Looking at figure 4.33 we can see that u1 (x, y) = 0 andu2 (x, y) = 1− x− y. The region D can be viewed as a type 1 region that is
D ={
(x, y) ∈ R2 : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1− x}
Hence ∫∫∫E
zdV =
∫ 1
0
∫ 1−x
0
∫ 1−x−y
0
zdzdydx
First, we evaluate the innermost integral.∫ 1−x−y
0
zdz =z2
2
∣∣∣∣1−x−y0
=(1− x− y)
2
2
4.7. TRIPLE INTEGRALS 393
Figure 4.34: Region D
394 CHAPTER 4. MULTIPLE INTEGRALS
The next inner integral is then∫ 1−x
0
∫ 1−x−y
0
zdzdy =1
2
∫ 1−x
0
(1− x− y) dy
=1
2
− (1− x− y)3
3
∣∣∣∣∣1−x
0
=
1
6
(−0 + (1− x)
3)
Finally ∫∫∫E
zdV =1
6
∫ 1
0
(1− x)3dx
=1
6
− (1− x)4
4
∣∣∣∣∣1
0
=
1
24(−0 + 1)
=1
24
4.7.5 Triple Integrals Over a Type 2 Solid
E is said to be a type 2 solid if it lies between the graphs of two functions of yand z in other words
E ={
(x, y, z) ∈ R3 : (x, y) ∈ D,u1 (y, z) ≤ x ≤ u2 (y, z)}
where D is the projection of E on the yz-plane. Such a region is shown on figure4.35.It can be proven that∫∫∫
E
f (x, y, z) dV =
∫∫D
[∫ u2(y,z)
u1(y,z)
f (x, y, z) dx
]dA
4.7.6 Triple Integrals Over a Type 3 Solid
E is said to be a type 3 solid if it lies between the graphs of two functions of xand z in other words
E ={
(x, y, z) ∈ R3 : (x, z) ∈ D,u1 (x, z) ≤ y ≤ u2 (x, z)}
where D is the projection of E on the xz-plane. Such a region is shown on figure4.36.
4.7. TRIPLE INTEGRALS 395
Figure 4.35: A Type 2 Solid
396 CHAPTER 4. MULTIPLE INTEGRALS
Figure 4.36: A Type 3 Solid
It can be proven that∫∫∫E
f (x, y, z) dV =
∫∫D
[∫ u2(x,z)
u1(x,z)
f (x, y, z) dy
]dA
Bibliography
[1] Joel Hass, Maurice D. Weir, and George B. Thomas, University calculus:Early transcendentals, Pearson Addison-Wesley, 2012.
[2] James Stewart, Calculus, Cengage Learning, 2011.
[3] Michael Sullivan and Kathleen Miranda, Calculus: Early transcendentals,Macmillan Higher Education, 2014.
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